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| Paper IPM / M / 177 |
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| Abstract: | |||||
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Among other results, it is shown that if C
and K are arbitrary complex n×n matrices and if det(λ02 I+ λ0 C+K)=0 for some λ0 ≠ 0 (resp. λ0=0), then the Newton diagram of the polynomial t(λ, ϵ)=det (λ2 I+λ(1+ϵ) C+K), expanded in (λ−λ0) and ϵ, has at least a point on or below the line x+y=b (
resp. has no point on or above the line x=y), where b is the algebraic multiplicity of 0 as an eigenvalue of
λ02I+λ0 C+K. These are extensions of similar results due to H. Langer, B. Najman, and K. Veseli\acutec proved for diagonable matrices C, and shed light on the eigenvalues of the perturbed quadratic matrix polynomials. Our proofs are independent and seem to be simpler.
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